Normal Distribution Cheatsheet

Cheatsheet Content

The Normal Distribution Also known as symmetric, bell-shaped distribution. Data is approximately normal. Characteristics: Symmetric about the mean, with a single peak. Mean = Median = Mode. 50% of data values are below the mean, and 50% are above. Data values farther from the mean become increasingly rare. Graph is bell-shaped with tails approaching, but never touching, the horizontal axis. Almost all area under the curve is within 3 standard deviations of the mean. Each tick mark on the horizontal axis marks increments of 1 standard deviation. The Empirical Rule (68-95-99.7 Rule) Describes the percentage of data values that fall within a certain number of standard deviations from the mean in a normal distribution. 68% of data values fall within 1 standard deviation of the mean ($ \mu \pm 1\sigma $). 95% of data values fall within 2 standard deviations of the mean ($ \mu \pm 2\sigma $). 99.7% of data values fall within 3 standard deviations of the mean ($ \mu \pm 3\sigma $). Mean -1$\sigma$ +1$\sigma$ -2$\sigma$ +2$\sigma$ -3$\sigma$ +3$\sigma$ Standard Scores (z-scores) A z-score measures the number of standard deviations a data value is from the mean. Used for the standard normal distribution , which has a population mean $\mu=0$ and standard deviation $\sigma=1$. Characteristics: $z=0$ corresponds to the mean of the distribution. If $z$ is positive, the data value is above the mean. If $z$ is negative, the data value is below the mean. Calculating z-score: $$ z = \frac{\text{data value} - \text{mean}}{\text{standard deviation}} = \frac{x - \mu}{\sigma} $$ Calculating data value from z-score: $$ x = z \times \text{standard deviation} + \text{mean} = z\sigma + \mu $$ Interpretation of z-scores (based on Empirical Rule): 68% of z-scores are between -1 and 1. 95% of z-scores are between -2 and 2. 99.7% of z-scores are between -3 and 3. A z-score below -3 or above 3 is possible but very unlikely. Finding Percentages with the Normal Distribution To find the percentage of observations (area) under the normal distribution curve: Calculate the z-score for the observed value $x$. Use a z-table or graphing calculator (e.g., normalcdf function) to find the area. Using a Graphing Calculator ( normalcdf ): Syntax: normalcdf(lower bound, upper bound, mean, standard deviation) For standard normal distribution , use $\mu=0$ and $\sigma=1$. For any normal distribution , use its specific $\mu$ and $\sigma$. Common Area Calculations: Area to the left of $z$ (or $x$): Calculator: normalcdf(-1000000, z, 0, 1) (for standard normal) Calculator: normalcdf(-1000000, x, $\mu$, $\sigma$) (for any normal) Area to the right of $z$ (or $x$): Calculator: normalcdf(z, 1000000, 0, 1) (for standard normal) Calculator: normalcdf(x, 1000000, $\mu$, $\sigma$) (for any normal) Alternatively: $1 - (\text{Area to the left of } z \text{ or } x)$ Area between two z-scores ($z_1$ and $z_2$) (or two data values $x_1$ and $x_2$): Calculator: normalcdf(z1, z2, 0, 1) (for standard normal) Calculator: normalcdf(x1, x2, $\mu$, $\sigma$) (for any normal) Alternatively: $(\text{Area to the left of } z_2) - (\text{Area to the left of } z_1)$ Note on extreme bounds: Use a very large negative number (e.g., -1,000,000) for "negative infinity" and a very large positive number (e.g., 1,000,000) for "positive infinity" as lower/upper bounds in normalcdf .